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Order of a permutation

A cycle that contains finitely many elements is a finite cycle and the cycle length is the number of elements it contains.

Let be a permutation written as a finite product of disjoint cycles of finite length. The order of is the least common multiple of the lengths of the cycles.

A transposition is a cycle of length two. Clearly a transposition has order two, but there are permutations of order two that are not transpositions.

One important application of transpositions is that every permutation may be written as a product of transpositions (although not necessarily disjoint and not uniquely). A permutation is an even permutation if it is a product of an even number of transpositions and an odd permutation if it is a product of an odd number of transpositions.

As the definition suggests, a permutation cannot be both odd and even. However, the decomposition into a product of transpositions is not unique nor is the number of transpositions unique. For example, the permutation may be written as , or . Of course, the identity permutation is an even permutation.

Peter Williams
Sun Mar 30 14:48:35 PST 1997